In [ ]:
Implement model for α-synuclein aggregation by Sneppen et al. presented in the lecture.¶
a. Start with the model with 7 parameters and through parameter rescaling arrive to 4-parameter model

b. In the presented version, protease production was assumed constant. How do the results change if you include the fact that protease is induced by

    i. Proto-fibrils (s)

    ii. Oligomers (ααα)

    iii. Fibrils, (F)

c. Which of i-iii have a maximal impact on model outcome?

d. How would you modify the model to include the effect of autophagy? How does this change the results?
In [1]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve_original_model(gamma=1,m=25,sigma=1,nu=1,
                         ic = [0,0,0], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
In [ ]:
sol_m25 = solve_original_model(m=25,tmax= 250)
sol_m10 = solve_original_model(m=10,tmax=220)
In [ ]:
plt.plot(sol_m25[0], sol_m25[1],label="F(t)",color="blue")
plt.plot(sol_m25[0],sol_m25[2],label="C(t)",color="green")
plt.plot(sol_m25[0],sol_m25[3],label="P(t)",color="red")

plt.xlim([50,220])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with m=25, $\nu = \gamma = \sigma =1$")
plt.show()
In [ ]:
plt.plot(sol_m10[0], sol_m10[1],label="F(t)",color="blue")
plt.plot(sol_m10[0],sol_m10[2],label="P(t)",color="red")
plt.plot(sol_m10[0],sol_m10[3],label="C(t)",color="green")
plt.xlim([50,220])
plt.ylim([0,100])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with m=10, $\nu = \gamma = \sigma =1$")
plt.show()
In [ ]:
def num_oligomers(sol):
  ts = sol[0]
  Ps = sol[3]

  cumulative_integral = [0] * len(ts)


  for i in range(2, len(ts)):
    area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
    cumulative_integral[i] = cumulative_integral[i-1] + area


  return([ts,cumulative_integral])
In [ ]:
# oligomers_m25 = num_oligomers(sol_m25)
oligomers_m10 = num_oligomers(sol_m10)

plt.plot(oligomers_m10[0],oligomers_m10[1],color="red")
plt.title(r"Oligomer accumultion $\left(\int 1/P dt\right)$ for $m=10$")
plt.xlabel("Time")
plt.ylabel("Oligomer accumulation")
plt.ylim([0,700])
plt.xlim([50,220])
# plt.plot(oligomers_m10[0],oligomers_m10[1])
Out[ ]:
(50.0, 220.0)
In [ ]:
In [ ]:
solve_original_model(m=1, sigma=1,tmax=2000,start_from=50)
Out[ ]:
[array([  50.49203535,   51.65935211,   52.91505153, ..., 1998.0728519 ,
        1999.24531165, 2000.        ]),
 array([0.50033912, 0.50019802, 0.50017659, ..., 0.50033534, 0.5002007 ,
        0.50003559]),
 array([0.49956204, 0.49974488, 0.49977235, ..., 0.4995669 , 0.49974139,
        0.49995525]),
 array([1.00071352, 1.00041571, 1.00037147, ..., 1.00070561, 1.00042142,
        1.00007325])]
In [ ]:
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))

for i,m in enumerate(np.logspace(0,2,mnum)):
  print(i)
  for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
    sol = solve_original_model(m=m, sigma=sigma,tmax=1000,start_from=50)
    oligomers = num_oligomers(sol)

    final_oligomer_number = oligomers[-1][-1]

    mat_oligomers[i,j] = final_oligomer_number

    print(m,sigma,final_oligomer_number)

0
1.0 0.1 9497.23837149094
1.0 0.11844845812380056 8021.484175784347
1.0 0.14030037231905737 6772.82796507447
1.0 0.16618362775387496 5719.918675913305
1.0 0.19684194472866123 4827.278163999395
1.0 0.23315624847200295 4073.776007186317
1.0 0.27616998133438475 3440.050476115269
1.0 0.32711908469136664 2904.3435956801495
1.0 0.38746751204561314 2450.472386433967
1.0 0.45894929374867993 2069.556378595097
1.0 0.5436183620153837 1745.6864258858702
1.0 0.6439075678850823 1474.1702801649012
1.0 0.7626985859023444 1242.1469311128121
1.0 0.9034047151333576 1048.9131739112513
1.0 1.070068955693175 885.2087410258573
1.0 1.2674801788800205 747.7423336190142
1.0 1.5013107289081735 631.5121204366185
1.0 1.7782794100389228 533.1107262598389
1.0 2.106344542324121 450.28709037723485
1.0 2.4949326331577466 380.03100385401854
1.0 2.955209235202888 320.8941670323715
1.0 3.5003997734299794 271.12725753267574
1.0 4.146169559796819 228.87169427931437
1.0 4.911073914777701 193.2457600442409
1.0 5.817091329374358 163.17146862474078
1.0 6.890254987297223 137.73274182899087
1.0 8.16140079325183 116.33197385472813
1.0 9.66705340091042 98.20727783562079
1.0 11.450475699382825 82.93051826543011
1.0 13.562911913759416 70.02227422561164
1.0 16.065060038537286 59.108827143845176
1.0 19.028815912310247 49.907055988040725
1.0 22.539339047347912 42.13679158144729
1.0 26.697499572879334 35.57868997687477
1.0 31.622776601683793 30.037945080059576
1
1.1450475699382818 0.1 9472.6406926014
1.1450475699382818 0.11844845812380056 8020.104980880068
1.1450475699382818 0.14030037231905737 6773.13887241559
1.1450475699382818 0.16618362775387496 5718.923516827068
1.1450475699382818 0.19684194472866123 4827.614771476256
1.1450475699382818 0.23315624847200295 4074.178115851987
1.1450475699382818 0.27616998133438475 3441.0499962178296
1.1450475699382818 0.32711908469136664 2904.390166389578
1.1450475699382818 0.38746751204561314 2452.2301992232874
1.1450475699382818 0.45894929374867993 2069.499047267869
1.1450475699382818 0.5436183620153837 1746.5416406612144
1.1450475699382818 0.6439075678850823 1470.684629634692
1.1450475699382818 0.7626985859023444 1242.4369129919817
1.1450475699382818 0.9034047151333576 1048.8555412886092
1.1450475699382818 1.070068955693175 885.9942617453254
1.1450475699382818 1.2674801788800205 747.2102092624503
1.1450475699382818 1.5013107289081735 631.0077076753848
1.1450475699382818 1.7782794100389228 533.2623207342565
1.1450475699382818 2.106344542324121 450.21421250030494
1.1450475699382818 2.4949326331577466 379.93121486741904
1.1450475699382818 2.955209235202888 321.0456446950777
1.1450475699382818 3.5003997734299794 270.9569559674931
1.1450475699382818 4.146169559796819 228.75627640390118
1.1450475699382818 4.911073914777701 193.28319151314247
1.1450475699382818 5.817091329374358 163.17858232217498
1.1450475699382818 6.890254987297223 137.7422335772176
1.1450475699382818 8.16140079325183 116.34498272346269
1.1450475699382818 9.66705340091042 98.21010329274442
1.1450475699382818 11.450475699382825 82.93192602625192
1.1450475699382818 13.562911913759416 70.02336998880936
1.1450475699382818 16.065060038537286 59.12189137971139
1.1450475699382818 19.028815912310247 49.9061643648849
1.1450475699382818 22.539339047347912 42.13703020780156
1.1450475699382818 26.697499572879334 35.579007766609756
1.1450475699382818 31.622776601683793 30.037711172367935
2
1.3111339374215643 0.1 9468.236077233752
1.3111339374215643 0.11844845812380056 7999.906441799619
1.3111339374215643 0.14030037231905737 6759.704998723282
1.3111339374215643 0.16618362775387496 5720.736571917915
1.3111339374215643 0.19684194472866123 4828.0459865016255
1.3111339374215643 0.23315624847200295 4075.5023071728597
1.3111339374215643 0.27616998133438475 3440.0882637877357
1.3111339374215643 0.32711908469136664 2904.2888899491777
1.3111339374215643 0.38746751204561314 2451.9089093628145
1.3111339374215643 0.45894929374867993 2068.785214535877
1.3111339374215643 0.5436183620153837 1746.2095714260156
1.3111339374215643 0.6439075678850823 1472.054526470731
1.3111339374215643 0.7626985859023444 1242.1309726311356
1.3111339374215643 0.9034047151333576 1049.1965214410081
1.3111339374215643 1.070068955693175 885.9550991481425
1.3111339374215643 1.2674801788800205 747.9904740467043
1.3111339374215643 1.5013107289081735 631.3033048464517
1.3111339374215643 1.7782794100389228 532.8171429631162
1.3111339374215643 2.106344542324121 450.2622247284063
1.3111339374215643 2.4949326331577466 380.2413177728331
1.3111339374215643 2.955209235202888 320.95186810264346
1.3111339374215643 3.5003997734299794 271.09228770241964
1.3111339374215643 4.146169559796819 228.81949577334532
1.3111339374215643 4.911073914777701 193.19683164551347
1.3111339374215643 5.817091329374358 163.20267110498108
1.3111339374215643 6.890254987297223 137.75706391156834
1.3111339374215643 8.16140079325183 116.3492027963671
1.3111339374215643 9.66705340091042 98.20919774028899
1.3111339374215643 11.450475699382825 82.9309825445315
1.3111339374215643 13.562911913759416 70.0075905625565
1.3111339374215643 16.065060038537286 59.110795816576186
1.3111339374215643 19.028815912310247 49.90678742063192
1.3111339374215643 22.539339047347912 42.13730866547052
1.3111339374215643 26.697499572879334 35.5789186460052
1.3111339374215643 31.622776601683793 30.03785230962289
3
1.5013107289081733 0.1 9463.341115800968
1.5013107289081733 0.11844845812380056 7995.833620470002
1.5013107289081733 0.14030037231905737 6756.333521503364
1.5013107289081733 0.16618362775387496 5707.871263045132
1.5013107289081733 0.19684194472866123 4818.484811806256
1.5013107289081733 0.23315624847200295 4077.056050058158
1.5013107289081733 0.27616998133438475 3440.7050949374716
1.5013107289081733 0.32711908469136664 2904.715377964601
1.5013107289081733 0.38746751204561314 2451.0897200183886
1.5013107289081733 0.45894929374867993 2069.7945456951315
1.5013107289081733 0.5436183620153837 1746.9442293253953
1.5013107289081733 0.6439075678850823 1471.6940993709745
1.5013107289081733 0.7626985859023444 1242.0959065255386
1.5013107289081733 0.9034047151333576 1048.5871515040108
1.5013107289081733 1.070068955693175 885.2729205043015
1.5013107289081733 1.2674801788800205 747.7292598767892
1.5013107289081733 1.5013107289081735 631.1031558707759
1.5013107289081733 1.7782794100389228 532.9804196242517
1.5013107289081733 2.106344542324121 450.2837921778958
1.5013107289081733 2.4949326331577466 380.24403467467306
1.5013107289081733 2.955209235202888 320.89665984213684
1.5013107289081733 3.5003997734299794 270.9881102697371
1.5013107289081733 4.146169559796819 228.871004704445
1.5013107289081733 4.911073914777701 193.2265620336789
1.5013107289081733 5.817091329374358 163.11628222469255
1.5013107289081733 6.890254987297223 137.77028714824826
1.5013107289081733 8.16140079325183 116.31071251681253
1.5013107289081733 9.66705340091042 98.21344364690877
1.5013107289081733 11.450475699382825 82.93651627206394
1.5013107289081733 13.562911913759416 70.01111744023284
1.5013107289081733 16.065060038537286 59.111371471614994
1.5013107289081733 19.028815912310247 49.90727418632101
1.5013107289081733 22.539339047347912 42.137721539846176
1.5013107289081733 26.697499572879334 35.57441495913946
1.5013107289081733 31.622776601683793 30.034564838810827
4
1.7190722018585745 0.1 9458.630866084206
1.7190722018585745 0.11844845812380056 7989.108206255075
1.7190722018585745 0.14030037231905737 6753.043116233515
1.7190722018585745 0.16618362775387496 5708.307197868557
1.7190722018585745 0.19684194472866123 4821.09204428659
1.7190722018585745 0.23315624847200295 4067.282537101664
1.7190722018585745 0.27616998133438475 3440.8878391878716
1.7190722018585745 0.32711908469136664 2904.4438788081466
1.7190722018585745 0.38746751204561314 2452.367097891485
1.7190722018585745 0.45894929374867993 2069.7820788509002
1.7190722018585745 0.5436183620153837 1747.0852772413007
1.7190722018585745 0.6439075678850823 1471.9303241697555
1.7190722018585745 0.7626985859023444 1242.853047118037
1.7190722018585745 0.9034047151333576 1048.357322550013
1.7190722018585745 1.070068955693175 885.4231198469593
1.7190722018585745 1.2674801788800205 747.8205525553444
1.7190722018585745 1.5013107289081735 631.6555554933819
1.7190722018585745 1.7782794100389228 532.714109138598
1.7190722018585745 2.106344542324121 449.8586107939892
1.7190722018585745 2.4949326331577466 380.2353476012989
1.7190722018585745 2.955209235202888 320.79882191942556
1.7190722018585745 3.5003997734299794 270.936315352451
1.7190722018585745 4.146169559796819 228.80512448100345
1.7190722018585745 4.911073914777701 193.26385070953643
1.7190722018585745 5.817091329374358 163.1474176676202
1.7190722018585745 6.890254987297223 137.7847305063572
1.7190722018585745 8.16140079325183 116.31826119483542
1.7190722018585745 9.66705340091042 98.21799700250781
1.7190722018585745 11.450475699382825 82.93873409155113
1.7190722018585745 13.562911913759416 70.01303331483957
1.7190722018585745 16.065060038537286 59.114649385144624
1.7190722018585745 19.028815912310247 49.908327666370575
1.7190722018585745 22.539339047347912 42.13731484526492
1.7190722018585745 26.697499572879334 35.57419257016327
1.7190722018585745 31.622776601683793 30.034863440680137
5
1.9684194472866121 0.1 9457.597368276682
1.9684194472866121 0.11844845812380056 7982.009377178454
1.9684194472866121 0.14030037231905737 6748.091461880111
1.9684194472866121 0.16618362775387496 5706.583023966229
1.9684194472866121 0.19684194472866123 4821.0909595606145
1.9684194472866121 0.23315624847200295 4069.473777920445
1.9684194472866121 0.27616998133438475 3434.208202127424
1.9684194472866121 0.32711908469136664 2905.149817843093
1.9684194472866121 0.38746751204561314 2452.8145233435325
1.9684194472866121 0.45894929374867993 2070.0318093898613
1.9684194472866121 0.5436183620153837 1743.8553966964103
1.9684194472866121 0.6439075678850823 1471.479234879925
1.9684194472866121 0.7626985859023444 1242.943478185857
1.9684194472866121 0.9034047151333576 1048.667202560259
1.9684194472866121 1.070068955693175 885.2891767918519
1.9684194472866121 1.2674801788800205 748.0711038583133
1.9684194472866121 1.5013107289081735 631.4870810408984
1.9684194472866121 1.7782794100389228 533.2304626022726
1.9684194472866121 2.106344542324121 450.16767408830054
1.9684194472866121 2.4949326331577466 380.280370494065
1.9684194472866121 2.955209235202888 320.8129706133555
1.9684194472866121 3.5003997734299794 271.098978250728
1.9684194472866121 4.146169559796819 228.87637255764602
1.9684194472866121 4.911073914777701 193.29471158551445
1.9684194472866121 5.817091329374358 163.17403605018973
1.9684194472866121 6.890254987297223 137.79393539956868
1.9684194472866121 8.16140079325183 116.32716023873058
1.9684194472866121 9.66705340091042 98.2173352276246
1.9684194472866121 11.450475699382825 82.93878846486825
1.9684194472866121 13.562911913759416 70.01433575248265
1.9684194472866121 16.065060038537286 59.115720832458095
1.9684194472866121 19.028815912310247 49.90883971489732
1.9684194472866121 22.539339047347912 42.13773372076552
1.9684194472866121 26.697499572879334 35.57948685718625
1.9684194472866121 31.622776601683793 30.035397219367347
6
2.2539339047347906 0.1 9465.126643789701
2.2539339047347906 0.11844845812380056 7979.020462996187
2.2539339047347906 0.14030037231905737 6742.10060700211
2.2539339047347906 0.16618362775387496 5703.312500689324
2.2539339047347906 0.19684194472866123 4822.666005756447
2.2539339047347906 0.23315624847200295 4071.618405767408
2.2539339047347906 0.27616998133438475 3434.7306237368316
2.2539339047347906 0.32711908469136664 2898.3945425477373
2.2539339047347906 0.38746751204561314 2446.841522978335
2.2539339047347906 0.45894929374867993 2069.7664205097544
2.2539339047347906 0.5436183620153837 1744.0555797006648
2.2539339047347906 0.6439075678850823 1471.737171992609
2.2539339047347906 0.7626985859023444 1242.2260115254978
2.2539339047347906 0.9034047151333576 1048.527041735428
2.2539339047347906 1.070068955693175 886.2078346357296
2.2539339047347906 1.2674801788800205 747.7765765564113
2.2539339047347906 1.5013107289081735 631.6413976135017
2.2539339047347906 1.7782794100389228 533.150376146884
2.2539339047347906 2.106344542324121 449.95231441890076
2.2539339047347906 2.4949326331577466 380.00092739671237
2.2539339047347906 2.955209235202888 320.8161359174143
2.2539339047347906 3.5003997734299794 271.0347478364891
2.2539339047347906 4.146169559796819 228.7722848698983
2.2539339047347906 4.911073914777701 193.20967515133168
2.2539339047347906 5.817091329374358 163.2064647660301
2.2539339047347906 6.890254987297223 137.75371552014792
2.2539339047347906 8.16140079325183 116.33176921804179
2.2539339047347906 9.66705340091042 98.22803217314635
2.2539339047347906 11.450475699382825 82.93879251387823
2.2539339047347906 13.562911913759416 70.01394878038903
2.2539339047347906 16.065060038537286 59.116610325126
2.2539339047347906 19.028815912310247 49.911077580519475
2.2539339047347906 22.539339047347912 42.139095891177014
2.2539339047347906 26.697499572879334 35.5748796736127
2.2539339047347906 31.622776601683793 30.035549024779584
7
2.580861540418075 0.1 9490.032880311614
2.580861540418075 0.11844845812380056 7982.480459690292
2.580861540418075 0.14030037231905737 6735.788729391186
2.580861540418075 0.16618362775387496 5696.9402516913
2.580861540418075 0.19684194472866123 4821.393532660229
2.580861540418075 0.23315624847200295 4072.512316914074
2.580861540418075 0.27616998133438475 3436.117489648797
2.580861540418075 0.32711908469136664 2898.23616187275
2.580861540418075 0.38746751204561314 2447.236718068838
2.580861540418075 0.45894929374867993 2065.5472669486944
2.580861540418075 0.5436183620153837 1744.5991469530122
2.580861540418075 0.6439075678850823 1472.4684654094028
2.580861540418075 0.7626985859023444 1242.6645718511325
2.580861540418075 0.9034047151333576 1048.944219790335
2.580861540418075 1.070068955693175 885.8348648022239
2.580861540418075 1.2674801788800205 747.6380972962759
2.580861540418075 1.5013107289081735 631.2178830147737
2.580861540418075 1.7782794100389228 533.1306526220769
2.580861540418075 2.106344542324121 450.33637464363557
2.580861540418075 2.4949326331577466 380.1503197361258
2.580861540418075 2.955209235202888 320.8946066530802
2.580861540418075 3.5003997734299794 271.03414949709617
2.580861540418075 4.146169559796819 228.93554126988494
2.580861540418075 4.911073914777701 193.27109173481495
2.580861540418075 5.817091329374358 163.15346659754874
2.580861540418075 6.890254987297223 137.77514531341566
2.580861540418075 8.16140079325183 116.34779820564972
2.580861540418075 9.66705340091042 98.23350437825708
2.580861540418075 11.450475699382825 82.91944177150799
2.580861540418075 13.562911913759416 70.01393956439408
2.580861540418075 16.065060038537286 59.11669959607663
2.580861540418075 19.028815912310247 49.91161849658742
2.580861540418075 22.539339047347912 42.13884072029476
2.580861540418075 26.697499572879334 35.576029980717976
2.580861540418075 31.622776601683793 30.03574832948225
8
2.955209235202887 0.1 9542.146280255924
2.955209235202887 0.11844845812380056 8002.159203914849
2.955209235202887 0.14030037231905737 6737.280467748147
2.955209235202887 0.16618362775387496 5691.167724835174
2.955209235202887 0.19684194472866123 4818.46432834495
2.955209235202887 0.23315624847200295 4075.9902816084464
2.955209235202887 0.27616998133438475 3438.9039845866364
2.955209235202887 0.32711908469136664 2899.013883216089
2.955209235202887 0.38746751204561314 2446.881484894785
2.955209235202887 0.45894929374867993 2066.117206637954
2.955209235202887 0.5436183620153837 1744.5404429928353
2.955209235202887 0.6439075678850823 1472.5246112133423
2.955209235202887 0.7626985859023444 1243.4146946490748
2.955209235202887 0.9034047151333576 1049.707413396152
2.955209235202887 1.070068955693175 885.872786468415
2.955209235202887 1.2674801788800205 747.8178284166834
2.955209235202887 1.5013107289081735 631.7344622673347
2.955209235202887 1.7782794100389228 533.2064375663807
2.955209235202887 2.106344542324121 450.29341958154384
2.955209235202887 2.4949326331577466 380.3232844934341
2.955209235202887 2.955209235202888 320.92687735277593
2.955209235202887 3.5003997734299794 271.0287608955949
2.955209235202887 4.146169559796819 228.88272770754995
2.955209235202887 4.911073914777701 193.21380279675463
2.955209235202887 5.817091329374358 163.19382107498564
2.955209235202887 6.890254987297223 137.7874589819319
2.955209235202887 8.16140079325183 116.31129562872852
2.955209235202887 9.66705340091042 98.23609930301137
2.955209235202887 11.450475699382825 82.92338562088491
2.955209235202887 13.562911913759416 70.01525470276478
2.955209235202887 16.065060038537286 59.11779586276893
2.955209235202887 19.028815912310247 49.91213577593578
2.955209235202887 22.539339047347912 42.140489495299484
2.955209235202887 26.697499572879334 35.57578603253609
2.955209235202887 31.622776601683793 30.03597539315943
9
3.383855153428234 0.1 9636.683417487415
3.383855153428234 0.11844845812380056 8046.775441897405
3.383855153428234 0.14030037231905737 6751.743898238039
3.383855153428234 0.16618362775387496 5690.796103517059
3.383855153428234 0.19684194472866123 4814.125112468294
3.383855153428234 0.23315624847200295 4077.0845825954034
3.383855153428234 0.27616998133438475 3441.7247163015
3.383855153428234 0.32711908469136664 2901.035153338642
3.383855153428234 0.38746751204561314 2446.587395042419
3.383855153428234 0.45894929374867993 2065.7460437762384
3.383855153428234 0.5436183620153837 1744.4642169671808
3.383855153428234 0.6439075678850823 1472.7494695477844
3.383855153428234 0.7626985859023444 1243.0498735643203
3.383855153428234 0.9034047151333576 1049.0179916980585
3.383855153428234 1.070068955693175 885.5590162721272
3.383855153428234 1.2674801788800205 747.6914716315777
3.383855153428234 1.5013107289081735 631.6903079119958
3.383855153428234 1.7782794100389228 532.9027340837739
3.383855153428234 2.106344542324121 450.2776364276511
3.383855153428234 2.4949326331577466 380.2647609575891
3.383855153428234 2.955209235202888 321.02902389069504
3.383855153428234 3.5003997734299794 271.0492194999927
3.383855153428234 4.146169559796819 228.8430518601909
3.383855153428234 4.911073914777701 193.30118674844152
3.383855153428234 5.817091329374358 163.12706804452029
3.383855153428234 6.890254987297223 137.75672590648776
3.383855153428234 8.16140079325183 116.32182718480328
3.383855153428234 9.66705340091042 98.21152888269968
3.383855153428234 11.450475699382825 82.92754730438863
3.383855153428234 13.562911913759416 70.02015053997556
3.383855153428234 16.065060038537286 59.1178155541753
3.383855153428234 19.028815912310247 49.91181756833814
3.383855153428234 22.539339047347912 42.140606191212136
3.383855153428234 26.697499572879334 35.57609765692424
3.383855153428234 31.622776601683793 30.03633012648382
10
3.874675120456131 0.1 9797.791249936801
3.874675120456131 0.11844845812380056 8133.057105665355
3.874675120456131 0.14030037231905737 6790.988004248459
3.874675120456131 0.16618362775387496 5702.500897735752
3.874675120456131 0.19684194472866123 4813.275987504911
3.874675120456131 0.23315624847200295 4076.534539770055
3.874675120456131 0.27616998133438475 3447.5776690054313
3.874675120456131 0.32711908469136664 2903.5329561676685
3.874675120456131 0.38746751204561314 2446.603082808738
3.874675120456131 0.45894929374867993 2065.743066735055
3.874675120456131 0.5436183620153837 1744.7744455249128
3.874675120456131 0.6439075678850823 1472.254564929014
3.874675120456131 0.7626985859023444 1243.3757305015026
3.874675120456131 0.9034047151333576 1049.7432981523139
3.874675120456131 1.070068955693175 886.0233944490944
3.874675120456131 1.2674801788800205 748.0480651476464
3.874675120456131 1.5013107289081735 631.3702955355013
3.874675120456131 1.7782794100389228 533.3974239433604
3.874675120456131 2.106344542324121 450.03620760640644
3.874675120456131 2.4949326331577466 380.22773688459415
3.874675120456131 2.955209235202888 320.9203573205378
3.874675120456131 3.5003997734299794 271.11562542982216
3.874675120456131 4.146169559796819 228.8642475708269
3.874675120456131 4.911073914777701 193.2673919341202
3.874675120456131 5.817091329374358 163.17691916768442
3.874675120456131 6.890254987297223 137.76832143598546
3.874675120456131 8.16140079325183 116.33603652225521
3.874675120456131 9.66705340091042 98.22264561386274
3.874675120456131 11.450475699382825 82.93148508038941
3.874675120456131 13.562911913759416 70.02152043564548
3.874675120456131 16.065060038537286 59.11924396290489
3.874675120456131 19.028815912310247 49.91149828675405
3.874675120456131 22.539339047347912 42.14008846319758
3.874675120456131 26.697499572879334 35.57639399594436
3.874675120456131 31.622776601683793 30.036531443594285
11
4.436687330978613 0.1 10058.833314789192
4.436687330978613 0.11844845812380056 8286.619152377902
4.436687330978613 0.14030037231905737 6874.418742133132
4.436687330978613 0.16618362775387496 5740.555961429353
4.436687330978613 0.19684194472866123 4824.634595780332
4.436687330978613 0.23315624847200295 4077.6437640230197
4.436687330978613 0.27616998133438475 3453.797374176267
4.436687330978613 0.32711908469136664 2911.0387518557786
4.436687330978613 0.38746751204561314 2448.519465748165
4.436687330978613 0.45894929374867993 2065.6561241988993
4.436687330978613 0.5436183620153837 1744.6532393361138
4.436687330978613 0.6439075678850823 1472.9343999861526
4.436687330978613 0.7626985859023444 1243.437951830087
4.436687330978613 0.9034047151333576 1049.3647802850267
4.436687330978613 1.070068955693175 886.2766911690458
4.436687330978613 1.2674801788800205 748.0681216772297
4.436687330978613 1.5013107289081735 631.4610186097459
4.436687330978613 1.7782794100389228 533.0093671415829
4.436687330978613 2.106344542324121 450.2924879600039
4.436687330978613 2.4949326331577466 379.9971317687743
4.436687330978613 2.955209235202888 321.11260147127695
4.436687330978613 3.5003997734299794 270.94211134241476
4.436687330978613 4.146169559796819 228.8800017406911
4.436687330978613 4.911073914777701 193.23689674064101
4.436687330978613 5.817091329374358 163.140448062179
4.436687330978613 6.890254987297223 137.7406735035246
4.436687330978613 8.16140079325183 116.30825632195537
4.436687330978613 9.66705340091042 98.23182987627389
4.436687330978613 11.450475699382825 82.93985829985756
4.436687330978613 13.562911913759416 70.02364667815658
4.436687330978613 16.065060038537286 59.119509235109504
4.436687330978613 19.028815912310247 49.91323938914881
4.436687330978613 22.539339047347912 42.14143852983967
4.436687330978613 26.697499572879334 35.57731736340469
4.436687330978613 31.622776601683793 30.036732161598
12
5.0802180469130205 0.1 10472.8289311093
5.0802180469130205 0.11844845812380056 8545.605497700346
5.0802180469130205 0.14030037231905737 7028.1156219954655
5.0802180469130205 0.16618362775387496 5825.075095312603
5.0802180469130205 0.19684194472866123 4864.664355169403
5.0802180469130205 0.23315624847200295 4091.7838161420964
5.0802180469130205 0.27616998133438475 3462.0501292412496
5.0802180469130205 0.32711908469136664 2923.500266884204
5.0802180469130205 0.38746751204561314 2453.9623029689665
5.0802180469130205 0.45894929374867993 2064.995414694062
5.0802180469130205 0.5436183620153837 1744.5874785578587
5.0802180469130205 0.6439075678850823 1473.2254591773312
5.0802180469130205 0.7626985859023444 1243.1223238393925
5.0802180469130205 0.9034047151333576 1049.6951972021689
5.0802180469130205 1.070068955693175 885.7222102755356
5.0802180469130205 1.2674801788800205 748.3121234647598
5.0802180469130205 1.5013107289081735 631.3825153728559
5.0802180469130205 1.7782794100389228 533.3996202314431
5.0802180469130205 2.106344542324121 450.3355100747949
5.0802180469130205 2.4949326331577466 380.13632099842454
5.0802180469130205 2.955209235202888 321.12354528435463
5.0802180469130205 3.5003997734299794 271.05860436474893
5.0802180469130205 4.146169559796819 228.91654627460588
5.0802180469130205 4.911073914777701 193.22587355589724
5.0802180469130205 5.817091329374358 163.20705199272248
5.0802180469130205 6.890254987297223 137.7695280069467
5.0802180469130205 8.16140079325183 116.31935277559556
5.0802180469130205 9.66705340091042 98.20580664408382
5.0802180469130205 11.450475699382825 82.94164597197747
5.0802180469130205 13.562911913759416 70.01019161304781
5.0802180469130205 16.065060038537286 59.10945314980301
5.0802180469130205 19.028815912310247 49.914314092588725
5.0802180469130205 22.539339047347912 42.14126743286577
5.0802180469130205 26.697499572879334 35.57734892487454
5.0802180469130205 31.622776601683793 30.036909926482053
13
5.817091329374358 0.1 11125.558457830166
5.817091329374358 0.11844845812380056 8974.649366790154
5.817091329374358 0.14030037231905737 7302.515291175464
5.817091329374358 0.16618362775387496 5992.190715499639
5.817091329374358 0.19684194472866123 4958.530703113695
5.817091329374358 0.23315624847200295 4137.72612582977
5.817091329374358 0.27616998133438475 3481.6255254965768
5.817091329374358 0.32711908469136664 2942.5541090415177
5.817091329374358 0.38746751204561314 2468.394208565973
5.817091329374358 0.45894929374867993 2068.2941450772823
5.817091329374358 0.5436183620153837 1744.1857874325956
5.817091329374358 0.6439075678850823 1474.0826075757998
5.817091329374358 0.7626985859023444 1242.9234886264978
5.817091329374358 0.9034047151333576 1049.6457680404233
5.817091329374358 1.070068955693175 886.2904133408595
5.817091329374358 1.2674801788800205 747.7825881900486
5.817091329374358 1.5013107289081735 631.4451316327863
5.817091329374358 1.7782794100389228 533.2383697163095
5.817091329374358 2.106344542324121 450.0480447681624
5.817091329374358 2.4949326331577466 380.01589021489184
5.817091329374358 2.955209235202888 320.93784370242633
5.817091329374358 3.5003997734299794 271.047184708139
5.817091329374358 4.146169559796819 228.8072414230789
5.817091329374358 4.911073914777701 193.2335193147359
5.817091329374358 5.817091329374358 163.17833956825373
5.817091329374358 6.890254987297223 137.79659523724757
5.817091329374358 8.16140079325183 116.3394700299194
5.817091329374358 9.66705340091042 98.21466046648861
5.817091329374358 11.450475699382825 82.922406473443
5.817091329374358 13.562911913759416 70.01187076744979
5.817091329374358 16.065060038537286 59.11139937977018
5.817091329374358 19.028815912310247 49.915583522928095
5.817091329374358 22.539339047347912 42.13476713096668
5.817091329374358 26.697499572879334 35.57744538281327
5.817091329374358 31.622776601683793 30.03743060159091
14
6.6608462908091575 0.1 12164.203688261188
6.6608462908091575 0.11844845812380056 9691.498607751237
6.6608462908091575 0.14030037231905737 7785.634391120844
6.6608462908091575 0.16618362775387496 6307.462504500562
6.6608462908091575 0.19684194472866123 5155.075936495892
6.6608462908091575 0.23315624847200295 4252.100174570863
6.6608462908091575 0.27616998133438475 3540.727758443801
6.6608462908091575 0.32711908469136664 2976.2770000370683
6.6608462908091575 0.38746751204561314 2498.0700326453366
6.6608462908091575 0.45894929374867993 2081.7730548763307
6.6608462908091575 0.5436183620153837 1745.1499152151505
6.6608462908091575 0.6439075678850823 1474.6349730018733
6.6608462908091575 0.7626985859023444 1243.8542301800521
6.6608462908091575 0.9034047151333576 1049.3771740596753
6.6608462908091575 1.070068955693175 885.8777493915915
6.6608462908091575 1.2674801788800205 748.0201931234277
6.6608462908091575 1.5013107289081735 631.7235816736069
6.6608462908091575 1.7782794100389228 533.3519775267252
6.6608462908091575 2.106344542324121 450.1802545033341
6.6608462908091575 2.4949326331577466 380.2360059652842
6.6608462908091575 2.955209235202888 321.0785847998368
6.6608462908091575 3.5003997734299794 271.0297046073945
6.6608462908091575 4.146169559796819 228.90254051167108
6.6608462908091575 4.911073914777701 193.26913450048073
6.6608462908091575 5.817091329374358 163.17301444961925
6.6608462908091575 6.890254987297223 137.77561776624373
6.6608462908091575 8.16140079325183 116.31372587195219
6.6608462908091575 9.66705340091042 98.2253829998211
6.6608462908091575 11.450475699382825 82.93150311839364
6.6608462908091575 13.562911913759416 70.01944513391355
6.6608462908091575 16.065060038537286 59.11393449368455
6.6608462908091575 19.028815912310247 49.90730660644959
6.6608462908091575 22.539339047347912 42.135553432468164
6.6608462908091575 26.697499572879334 35.5773446926798
6.6608462908091575 31.622776601683793 30.037682034490167
15
7.626985859023444 0.1 13919.66961540925
7.626985859023444 0.11844845812380056 10931.941482931263
7.626985859023444 0.14030037231905737 8647.600055490984
7.626985859023444 0.16618362775387496 6907.104841161707
7.626985859023444 0.19684194472866123 5560.846161336459
7.626985859023444 0.23315624847200295 4516.850619581247
7.626985859023444 0.27616998133438475 3705.093543333052
7.626985859023444 0.32711908469136664 3069.2483376590667
7.626985859023444 0.38746751204561314 2563.2187802434664
7.626985859023444 0.45894929374867993 2120.7940631889132
7.626985859023444 0.5436183620153837 1757.8205610271743
7.626985859023444 0.6439075678850823 1475.9841652044843
7.626985859023444 0.7626985859023444 1245.273393272473
7.626985859023444 0.9034047151333576 1049.3415198407104
7.626985859023444 1.070068955693175 886.2677615711004
7.626985859023444 1.2674801788800205 748.238769016467
7.626985859023444 1.5013107289081735 631.4733439983296
7.626985859023444 1.7782794100389228 533.2539092516934
7.626985859023444 2.106344542324121 450.24136924534173
7.626985859023444 2.4949326331577466 380.31091371151047
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16
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17
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18
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19
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20
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22
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24
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87.33261623828433 0.1 586476.9901977802
87.33261623828433 0.11844845812380056 456445.9541731437
87.33261623828433 0.14030037231905737 346636.153486181
87.33261623828433 0.16618362775387496 253889.2054792302
87.33261623828433 0.19684194472866123 193839.10916512876
87.33261623828433 0.23315624847200295 165434.56568491852
87.33261623828433 0.27616998133438475 134589.45249835952
87.33261623828433 0.32711908469136664 100324.85694102883
87.33261623828433 0.38746751204561314 83986.24966897808
87.33261623828433 0.45894929374867993 67565.53961598183
87.33261623828433 0.5436183620153837 54791.46840069888
87.33261623828433 0.6439075678850823 42619.55593176349
87.33261623828433 0.7626985859023444 35811.286095909076
87.33261623828433 0.9034047151333576 29213.74510676359
87.33261623828433 1.070068955693175 23362.774309645356
87.33261623828433 1.2674801788800205 19072.149167622356
87.33261623828433 1.5013107289081735 15278.909758905893
87.33261623828433 1.7782794100389228 12283.213747295158
87.33261623828433 2.106344542324121 9851.69378829054
87.33261623828433 2.4949326331577466 7719.950424500689
87.33261623828433 2.955209235202888 5923.479812012361
87.33261623828433 3.5003997734299794 4451.560086226327
87.33261623828433 4.146169559796819 3059.740735388424
87.33261623828433 4.911073914777701 1777.7216635567547
87.33261623828433 5.817091329374358 163.56573657553653
87.33261623828433 6.890254987297223 137.7595296059756
87.33261623828433 8.16140079325183 116.31478468199533
87.33261623828433 9.66705340091042 98.23138123953139
87.33261623828433 11.450475699382825 82.92220532874191
87.33261623828433 13.562911913759416 70.02389225258807
87.33261623828433 16.065060038537286 59.11633197285232
87.33261623828433 19.028815912310247 49.91041077077265
87.33261623828433 22.539339047347912 42.140242562570144
87.33261623828433 26.697499572879334 35.57443940044335
87.33261623828433 31.622776601683793 30.03630185520499
34
100.0 0.1 705923.7676570666
100.0 0.11844845812380056 557279.2371946435
100.0 0.14030037231905737 431745.62433804997
100.0 0.16618362775387496 325728.42626543297
100.0 0.19684194472866123 236157.8072892566
100.0 0.23315624847200295 188323.00931877817
100.0 0.27616998133438475 160274.23495368764
In [ ]:
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()
ax = plt.axes(111, projection='3d')

ax.set_yscale("log")
ax.set_xscale("log")
# ax.set_zscale("log")

x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)

ax.plot_wireframe(x,y, np.log(mat_oligomers.T))


ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('ln(oligomers)')
ax.view_init(elev=30, azim=120)

plt.show()
In [ ]:
In [ ]:

Protease can be induced by proto fibrils¶

In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta*m/(1+P)

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])


def num_oligomers(sol):
  ts = sol[0]
  Ps = sol[3]

  cumulative_integral = [0] * len(ts)


  for i in range(2, len(ts)):
    area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
    cumulative_integral[i] = cumulative_integral[i-1] + area


  return([ts,cumulative_integral])
In [ ]:
sol = solve(m=5000,beta=0.4,tmax=5000)
In [ ]:
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([4500,5000])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"$\beta = 0.4$ and $m = 5000$")
plt.show()
In [ ]:
oligomers = num_oligomers(sol)

plt.plot(oligomers[0],oligomers[1],color="red")
Out[ ]:
[<matplotlib.lines.Line2D at 0x7fe1d5a638e0>]
In [ ]:
beta = 5
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))

for i,m in enumerate(np.logspace(0,2,mnum)):
  print(i)
  for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
    sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
    oligomers = num_oligomers(sol)

    final_oligomer_number = oligomers[-1][-1]

    mat_oligomers[i,j] = final_oligomer_number

    print(m,sigma,final_oligomer_number)

0
1.0 0.1 511.41766281010337
1.0 0.11844845812380056 508.282307598556
1.0 0.14030037231905737 504.6094489466907
1.0 0.16618362775387496 500.3149069386491
1.0 0.19684194472866123 495.30446272587585
1.0 0.23315624847200295 489.47253410520057
1.0 0.27616998133438475 482.7046971936214
1.0 0.32711908469136664 474.82948603786764
1.0 0.38746751204561314 465.9060063600262
1.0 0.45894929374867993 455.5689395363075
1.0 0.5436183620153837 443.7709516259087
1.0 0.6439075678850823 429.95827983023764
1.0 0.7626985859023444 415.05735626377776
1.0 0.9034047151333576 397.93644606713445
1.0 1.070068955693175 379.3205528228079
1.0 1.2674801788800205 359.0752990157088
1.0 1.5013107289081735 337.05017398350293
1.0 1.7782794100389228 313.80658829990904
1.0 2.106344542324121 289.4911600128019
1.0 2.4949326331577466 264.7078347305417
1.0 2.955209235202888 239.32259890734198
1.0 3.5003997734299794 214.5604295293716
1.0 4.146169559796819 190.46533457148684
1.0 4.911073914777701 167.68565500827373
1.0 5.817091329374358 146.39611443138693
1.0 6.890254987297223 126.91870854305198
1.0 8.16140079325183 109.3999702720464
1.0 9.66705340091042 93.86026440117685
1.0 11.450475699382825 80.18721126713496
1.0 13.562911913759416 68.32501650281323
1.0 16.065060038537286 58.07714643637219
1.0 19.028815912310247 49.2682788563647
1.0 22.539339047347912 41.74960100979572
1.0 26.697499572879334 35.340518737938105
1.0 31.622776601683793 29.891223943102787
1
1.1450475699382818 0.1 472.63868887726437
1.1450475699382818 0.11844845812380056 469.99960666938983
1.1450475699382818 0.14030037231905737 466.9029897820838
1.1450475699382818 0.16618362775387496 463.2755525575713
1.1450475699382818 0.19684194472866123 459.035570239308
1.1450475699382818 0.23315624847200295 454.0908235594041
1.1450475699382818 0.27616998133438475 448.3410275073302
1.1450475699382818 0.32711908469136664 441.58914820654854
1.1450475699382818 0.38746751204561314 433.8969072868071
1.1450475699382818 0.45894929374867993 425.1155464603425
1.1450475699382818 0.5436183620153837 414.5316522492409
1.1450475699382818 0.6439075678850823 403.08482376934745
1.1450475699382818 0.7626985859023444 390.163110231791
1.1450475699382818 0.9034047151333576 375.23705480869137
1.1450475699382818 1.070068955693175 358.90707067502365
1.1450475699382818 1.2674801788800205 341.0865479399444
1.1450475699382818 1.5013107289081735 321.4351051912131
1.1450475699382818 1.7782794100389228 300.5502666442933
1.1450475699382818 2.106344542324121 278.4995537248994
1.1450475699382818 2.4949326331577466 255.60337938750845
1.1450475699382818 2.955209235202888 232.37340588479088
1.1450475699382818 3.5003997734299794 209.26343380290646
1.1450475699382818 4.146169559796819 186.51587298984822
1.1450475699382818 4.911073914777701 164.84861738610914
1.1450475699382818 5.817091329374358 144.39894300387053
1.1450475699382818 6.890254987297223 125.60323562960279
1.1450475699382818 8.16140079325183 108.51752696763496
1.1450475699382818 9.66705340091042 93.28644904060269
1.1450475699382818 11.450475699382825 79.82022424129434
1.1450475699382818 13.562911913759416 68.09176938168913
1.1450475699382818 16.065060038537286 57.92986760944472
1.1450475699382818 19.028815912310247 49.18310216435592
1.1450475699382818 22.539339047347912 41.69150046738988
1.1450475699382818 26.697499572879334 35.30445543406245
1.1450475699382818 31.622776601683793 29.872058290294298
2
1.3111339374215643 0.1 437.00205505118583
1.3111339374215643 0.11844845812380056 434.7779241014333
1.3111339374215643 0.14030037231905737 432.1652068208496
1.3111339374215643 0.16618362775387496 429.10175332113323
1.3111339374215643 0.19684194472866123 425.5161997614192
1.3111339374215643 0.23315624847200295 421.3281872320487
1.3111339374215643 0.27616998133438475 416.44940063003645
1.3111339374215643 0.32711908469136664 410.7994126985323
1.3111339374215643 0.38746751204561314 404.26150956564646
1.3111339374215643 0.45894929374867993 396.65722570099297
1.3111339374215643 0.5436183620153837 387.59000904269146
1.3111339374215643 0.6439075678850823 377.7125793978103
1.3111339374215643 0.7626985859023444 366.14387999820235
1.3111339374215643 0.9034047151333576 353.5784551335615
1.3111339374215643 1.070068955693175 339.24588081707947
1.3111339374215643 1.2674801788800205 323.1769552043228
1.3111339374215643 1.5013107289081735 306.032146256447
1.3111339374215643 1.7782794100389228 287.41150989489796
1.3111339374215643 2.106344542324121 267.43960766448987
1.3111339374215643 2.4949326331577466 246.56270719312514
1.3111339374215643 2.955209235202888 225.19940013613495
1.3111339374215643 3.5003997734299794 203.69778502973054
1.3111339374215643 4.146169559796819 182.35165083117815
1.3111339374215643 4.911073914777701 161.81540503071346
1.3111339374215643 5.817091329374358 142.25932883535677
1.3111339374215643 6.890254987297223 124.10882352647442
1.3111339374215643 8.16140079325183 107.55205675711997
1.3111339374215643 9.66705340091042 92.64730507684008
1.3111339374215643 11.450475699382825 79.41074472507066
1.3111339374215643 13.562911913759416 67.82804595561943
1.3111339374215643 16.065060038537286 57.76185578895201
1.3111339374215643 19.028815912310247 49.078465498712596
1.3111339374215643 22.539339047347912 41.631787021014595
1.3111339374215643 26.697499572879334 35.26831563981717
1.3111339374215643 31.622776601683793 29.847383432197837
3
1.5013107289081733 0.1 404.4581683016634
1.5013107289081733 0.11844845812380056 402.579427442462
1.5013107289081733 0.14030037231905737 400.3712162787649
1.5013107289081733 0.16618362775387496 397.77905270320406
1.5013107289081733 0.19684194472866123 394.7410757923733
1.5013107289081733 0.23315624847200295 391.1875621833229
1.5013107289081733 0.27616998133438475 387.04144647120336
1.5013107289081733 0.32711908469136664 382.1278092075255
1.5013107289081733 0.38746751204561314 376.2007087161345
1.5013107289081733 0.45894929374867993 369.720739301583
1.5013107289081733 0.5436183620153837 362.2626530031501
1.5013107289081733 0.6439075678850823 353.3939759212841
1.5013107289081733 0.7626985859023444 343.7385958904093
1.5013107289081733 0.9034047151333576 332.4805069595212
1.5013107289081733 1.070068955693175 319.97618018584944
1.5013107289081733 1.2674801788800205 306.16112499857724
1.5013107289081733 1.5013107289081735 290.77453310831436
1.5013107289081733 1.7782794100389228 274.09743036942433
1.5013107289081733 2.106344542324121 256.21040655767115
1.5013107289081733 2.4949326331577466 237.4458217790953
1.5013107289081733 2.955209235202888 217.80949288210084
1.5013107289081733 3.5003997734299794 197.9159276101482
1.5013107289081733 4.146169559796819 177.99043578969213
1.5013107289081733 4.911073914777701 158.5755086308655
1.5013107289081733 5.817091329374358 139.93999792297643
1.5013107289081733 6.890254987297223 122.49820789071302
1.5013107289081733 8.16140079325183 106.44412061021633
1.5013107289081733 9.66705340091042 91.91046847194896
1.5013107289081733 11.450475699382825 78.94994703892694
1.5013107289081733 13.562911913759416 67.53521334080516
1.5013107289081733 16.065060038537286 57.57476926265253
1.5013107289081733 19.028815912310247 48.962389976208684
1.5013107289081733 22.539339047347912 41.558218997592704
1.5013107289081733 26.697499572879334 35.2222446908222
1.5013107289081733 31.622776601683793 29.822163847588484
4
1.7190722018585745 0.1 374.5511140806715
1.7190722018585745 0.11844845812380056 372.9566884676533
1.7190722018585745 0.14030037231905737 370.6964721369246
1.7190722018585745 0.16618362775387496 368.49801101013804
1.7190722018585745 0.19684194472866123 365.91937415425497
1.7190722018585745 0.23315624847200295 362.90019598482303
1.7190722018585745 0.27616998133438475 359.42390447127633
1.7190722018585745 0.32711908469136664 355.2767565087485
1.7190722018585745 0.38746751204561314 350.50736574954186
1.7190722018585745 0.45894929374867993 344.9643169298089
1.7190722018585745 0.5436183620153837 338.23643357172523
1.7190722018585745 0.6439075678850823 330.9283654361597
1.7190722018585745 0.7626985859023444 322.28375402369767
1.7190722018585745 0.9034047151333576 312.5359678710164
1.7190722018585745 1.070068955693175 301.60413706989107
1.7190722018585745 1.2674801788800205 289.490107401534
1.7190722018585745 1.5013107289081735 275.9044163415283
1.7190722018585745 1.7782794100389228 261.08830193019077
1.7190722018585745 2.106344542324121 245.05307778010578
1.7190722018585745 2.4949326331577466 228.09838480658033
1.7190722018585745 2.955209235202888 210.1865171365714
1.7190722018585745 3.5003997734299794 191.85229142767952
1.7190722018585745 4.146169559796819 173.4052666002215
1.7190722018585745 4.911073914777701 155.1698532640801
1.7190722018585745 5.817091329374358 137.454572589846
1.7190722018585745 6.890254987297223 120.80617306715513
1.7190722018585745 8.16140079325183 105.2832388446823
1.7190722018585745 9.66705340091042 91.12664512436503
1.7190722018585745 11.450475699382825 78.43711993112973
1.7190722018585745 13.562911913759416 67.208200909167
1.7190722018585745 16.065060038537286 57.36707873619565
1.7190722018585745 19.028815912310247 48.8299077947072
1.7190722018585745 22.539339047347912 41.47606829179049
1.7190722018585745 26.697499572879334 35.17048048790467
1.7190722018585745 31.622776601683793 29.790118805901052
5
1.9684194472866121 0.1 346.80244750775967
1.9684194472866121 0.11844845812380056 345.45516462696764
1.9684194472866121 0.14030037231905737 343.82046752169185
1.9684194472866121 0.16618362775387496 341.95538466470674
1.9684194472866121 0.19684194472866123 339.76565485382
1.9684194472866121 0.23315624847200295 337.19880371358164
1.9684194472866121 0.27616998133438475 333.8706464655501
1.9684194472866121 0.32711908469136664 330.4057373249132
1.9684194472866121 0.38746751204561314 326.3321180578828
1.9684194472866121 0.45894929374867993 321.6036607916845
1.9684194472866121 0.5436183620153837 315.82433538278343
1.9684194472866121 0.6439075678850823 309.53422174109227
1.9684194472866121 0.7626985859023444 302.0160250603347
1.9684194472866121 0.9034047151333576 293.53031278283424
1.9684194472866121 1.070068955693175 284.0279932773085
1.9684194472866121 1.2674801788800205 273.4199999354075
1.9684194472866121 1.5013107289081735 261.4657363545634
1.9684194472866121 1.7782794100389228 248.3723074916125
1.9684194472866121 2.106344542324121 233.99180436447497
1.9684194472866121 2.4949326331577466 218.74969143333882
1.9684194472866121 2.955209235202888 202.44874012734437
1.9684194472866121 3.5003997734299794 185.73095020825866
1.9684194472866121 4.146169559796819 168.52982718159146
1.9684194472866121 4.911073914777701 151.47943511356442
1.9684194472866121 5.817091329374358 134.81375301934372
1.9684194472866121 6.890254987297223 118.91614764856253
1.9684194472866121 8.16140079325183 103.97502627583317
1.9684194472866121 9.66705340091042 90.24016635695675
1.9684194472866121 11.450475699382825 77.84623269556951
1.9684194472866121 13.562911913759416 66.837217503968
1.9684194472866121 16.065060038537286 57.13251514679472
1.9684194472866121 19.028815912310247 48.681876365155105
1.9684194472866121 22.539339047347912 41.38202701182096
1.9684194472866121 26.697499572879334 35.11704609826163
1.9684194472866121 31.622776601683793 29.756784176515428
6
2.2539339047347906 0.1 321.33776646569055
2.2539339047347906 0.11844845812380056 320.1942918459013
2.2539339047347906 0.14030037231905737 318.84750472258264
2.2539339047347906 0.16618362775387496 317.2629691666886
2.2539339047347906 0.19684194472866123 315.4010630605729
2.2539339047347906 0.23315624847200295 313.2162775577038
2.2539339047347906 0.27616998133438475 310.34435631117384
2.2539339047347906 0.32711908469136664 307.3595347630777
2.2539339047347906 0.38746751204561314 303.8783923873991
2.2539339047347906 0.45894929374867993 299.82973588717397
2.2539339047347906 0.5436183620153837 294.84635155288447
2.2539339047347906 0.6439075678850823 289.4343150836963
2.2539339047347906 0.7626985859023444 282.96454929602584
2.2539339047347906 0.9034047151333576 275.63963136394074
2.2539339047347906 1.070068955693175 267.3570824606279
2.2539339047347906 1.2674801788800205 258.06550926803015
2.2539339047347906 1.5013107289081735 247.54542344318904
2.2539339047347906 1.7782794100389228 235.96359167633125
2.2539339047347906 2.106344542324121 223.13813897573075
2.2539339047347906 2.4949326331577466 209.2943000714285
2.2539339047347906 2.955209235202888 194.6856708662115
2.2539339047347906 3.5003997734299794 179.39614104695963
2.2539339047347906 4.146169559796819 163.62092673404834
2.2539339047347906 4.911073914777701 147.70545774529637
2.2539339047347906 5.817091329374358 132.0085100948468
2.2539339047347906 6.890254987297223 116.83777375748848
2.2539339047347906 8.16140079325183 102.55574651610924
2.2539339047347906 9.66705340091042 89.26562623427624
2.2539339047347906 11.450475699382825 77.21934103569886
2.2539339047347906 13.562911913759416 66.4071478676469
2.2539339047347906 16.065060038537286 56.869311751691846
2.2539339047347906 19.028815912310247 48.51407894582479
2.2539339047347906 22.539339047347912 41.278102890162266
2.2539339047347906 26.697499572879334 35.051433291214
2.2539339047347906 31.622776601683793 29.716317228979023
7
2.580861540418075 0.1 297.9471102095546
2.580861540418075 0.11844845812380056 296.9754228876133
2.580861540418075 0.14030037231905737 295.83037484680887
2.580861540418075 0.16618362775387496 294.4822661740634
2.580861540418075 0.19684194472866123 292.65534695329376
2.580861540418075 0.23315624847200295 290.7756201364797
2.580861540418075 0.27616998133438475 288.5972094088539
2.580861540418075 0.32711908469136664 286.04718997050406
2.580861540418075 0.38746751204561314 283.0687196459785
2.580861540418075 0.45894929374867993 279.37218635517036
2.580861540418075 0.5436183620153837 275.3458171187953
2.580861540418075 0.6439075678850823 270.4478761700598
2.580861540418075 0.7626985859023444 265.08032868702315
2.580861540418075 0.9034047151333576 258.6872170707294
2.580861540418075 1.070068955693175 251.47695291415522
2.580861540418075 1.2674801788800205 243.17015496202382
2.580861540418075 1.5013107289081735 233.93132444718142
2.580861540418075 1.7782794100389228 223.7249868578799
2.580861540418075 2.106344542324121 212.49069689227994
2.580861540418075 2.4949326331577466 200.07495950586886
2.580861540418075 2.955209235202888 186.8934116442059
2.580861540418075 3.5003997734299794 172.85012733968696
2.580861540418075 4.146169559796819 158.443314255163
2.580861540418075 4.911073914777701 143.68319623471467
2.580861540418075 5.817091329374358 128.98892764380065
2.580861540418075 6.890254987297223 114.62567683768107
2.580861540418075 8.16140079325183 100.98044819698436
2.580861540418075 9.66705340091042 88.20885109398272
2.580861540418075 11.450475699382825 76.5025198508123
2.580861540418075 13.562911913759416 65.95110411697013
2.580861540418075 16.065060038537286 56.559882524479164
2.580861540418075 19.028815912310247 48.32713743981776
2.580861540418075 22.539339047347912 41.16012534880641
2.580861540418075 26.697499572879334 34.97756325128403
2.580861540418075 31.622776601683793 29.670497535168423
8
2.955209235202887 0.1 276.2478820709047
2.955209235202887 0.11844845812380056 275.42155468073696
2.955209235202887 0.14030037231905737 274.4266563109408
2.955209235202887 0.16618362775387496 273.27892943079195
2.955209235202887 0.19684194472866123 271.951416416868
2.955209235202887 0.23315624847200295 270.34534713709303
2.955209235202887 0.27616998133438475 268.48397371702816
2.955209235202887 0.32711908469136664 266.0970147455213
2.955209235202887 0.38746751204561314 263.5484826607161
2.955209235202887 0.45894929374867993 260.569922440526
2.955209235202887 0.5436183620153837 256.88797804039143
2.955209235202887 0.6439075678850823 252.85818315999518
2.955209235202887 0.7626985859023444 248.03018354119934
2.955209235202887 0.9034047151333576 242.52312900329375
2.955209235202887 1.070068955693175 236.27053731169852
2.955209235202887 1.2674801788800205 229.17551699447634
2.955209235202887 1.5013107289081735 221.0767538782186
2.955209235202887 1.7782794100389228 212.03964703493907
2.955209235202887 2.106344542324121 201.9570062021607
2.955209235202887 2.4949326331577466 190.87632149812947
2.955209235202887 2.955209235202888 179.03033912548196
2.955209235202887 3.5003997734299794 166.4089754664992
2.955209235202887 4.146169559796819 153.12084623218405
2.955209235202887 4.911073914777701 139.46785820249102
2.955209235202887 5.817091329374358 125.8199966412859
2.955209235202887 6.890254987297223 112.28507580815855
2.955209235202887 8.16140079325183 99.32530471648539
2.955209235202887 9.66705340091042 87.0568908973943
2.955209235202887 11.450475699382825 75.71719520629303
2.955209235202887 13.562911913759416 65.43084656609727
2.955209235202887 16.065060038537286 56.23093317271097
2.955209235202887 19.028815912310247 48.107145160287736
2.955209235202887 22.539339047347912 41.02706517867638
2.955209235202887 26.697499572879334 34.89420426566967
2.955209235202887 31.622776601683793 29.618299341581253
9
3.383855153428234 0.1 256.4930826401304
3.383855153428234 0.11844845812380056 255.59894319758018
3.383855153428234 0.14030037231905737 254.7679017187442
3.383855153428234 0.16618362775387496 253.7892243734879
3.383855153428234 0.19684194472866123 252.62202479018137
3.383855153428234 0.23315624847200295 251.2588825281351
3.383855153428234 0.27616998133438475 249.67380647821273
3.383855153428234 0.32711908469136664 247.80758788390222
3.383855153428234 0.38746751204561314 245.44543145325414
3.383855153428234 0.45894929374867993 242.8716725299767
3.383855153428234 0.5436183620153837 239.7287989038192
3.383855153428234 0.6439075678850823 236.26458850292136
3.383855153428234 0.7626985859023444 232.10317848475194
3.383855153428234 0.9034047151333576 227.318629523202
3.383855153428234 1.070068955693175 221.88121508367158
3.383855153428234 1.2674801788800205 215.69470077277433
3.383855153428234 1.5013107289081735 208.58440354401918
3.383855153428234 1.7782794100389228 200.6441741134186
3.383855153428234 2.106344542324121 191.74444965244192
3.383855153428234 2.4949326331577466 181.97367000925433
3.383855153428234 2.955209235202888 171.22279755953588
3.383855153428234 3.5003997734299794 159.82582963548495
3.383855153428234 4.146169559796819 147.77239913439672
3.383855153428234 4.911073914777701 135.2093269725531
3.383855153428234 5.817091329374358 122.51286582631934
3.383855153428234 6.890254987297223 109.8525131567088
3.383855153428234 8.16140079325183 97.52124781723711
3.383855153428234 9.66705340091042 85.77936146857495
3.383855153428234 11.450475699382825 74.84185401490208
3.383855153428234 13.562911913759416 64.8528060029349
3.383855153428234 16.065060038537286 55.8529320592251
3.383855153428234 19.028815912310247 47.87144006102196
3.383855153428234 22.539339047347912 40.8713424959513
3.383855153428234 26.697499572879334 34.79649448485387
3.383855153428234 31.622776601683793 29.559774774607646
10
3.874675120456131 0.1 238.12912081169748
3.874675120456131 0.11844845812380056 237.52789404886965
3.874675120456131 0.14030037231905737 236.65403186748014
3.874675120456131 0.16618362775387496 235.79684571756678
3.874675120456131 0.19684194472866123 234.8134372210066
3.874675120456131 0.23315624847200295 233.65582933469847
3.874675120456131 0.27616998133438475 232.29468653735628
3.874675120456131 0.32711908469136664 230.6963463457469
3.874675120456131 0.38746751204561314 228.66235768565102
3.874675120456131 0.45894929374867993 226.4702425150577
3.874675120456131 0.5436183620153837 223.740167603359
3.874675120456131 0.6439075678850823 220.76192613722296
3.874675120456131 0.7626985859023444 217.14558520210448
3.874675120456131 0.9034047151333576 213.01261204940081
3.874675120456131 1.070068955693175 208.29045879698623
3.874675120456131 1.2674801788800205 202.76904813409126
3.874675120456131 1.5013107289081735 196.67634958374072
3.874675120456131 1.7782794100389228 189.57959776922615
3.874675120456131 2.106344542324121 181.82856826929046
3.874675120456131 2.4949326331577466 173.06908486809849
3.874675120456131 2.955209235202888 163.55480462138595
3.874675120456131 3.5003997734299794 153.2529485580325
3.874675120456131 4.146169559796819 142.2467876860025
3.874675120456131 4.911073914777701 130.78261769716525
3.874675120456131 5.817091329374358 119.03749801142612
3.874675120456131 6.890254987297223 107.20304453603887
3.874675120456131 8.16140079325183 95.57638389344557
3.874675120456131 9.66705340091042 84.40775445593506
3.874675120456131 11.450475699382825 73.90908304897638
3.874675120456131 13.562911913759416 64.20640426248413
3.874675120456131 16.065060038537286 55.435723542218355
3.874675120456131 19.028815912310247 47.59945551436414
3.874675120456131 22.539339047347912 40.70301924325954
3.874675120456131 26.697499572879334 34.6914909261635
3.874675120456131 31.622776601683793 29.49116091571142
11
4.436687330978613 0.1 221.06219630370356
4.436687330978613 0.11844845812380056 220.54852526836754
4.436687330978613 0.14030037231905737 219.9241474516388
4.436687330978613 0.16618362775387496 219.20989712719017
4.436687330978613 0.19684194472866123 218.36788123363368
4.436687330978613 0.23315624847200295 217.37607085915712
4.436687330978613 0.27616998133438475 216.06904392826507
4.436687330978613 0.32711908469136664 214.69887660519063
4.436687330978613 0.38746751204561314 213.09094934468547
4.436687330978613 0.45894929374867993 211.05903542925716
4.436687330978613 0.5436183620153837 208.85783117926738
4.436687330978613 0.6439075678850823 206.14564410690176
4.436687330978613 0.7626985859023444 203.0250405744513
4.436687330978613 0.9034047151333576 199.4298981576248
4.436687330978613 1.070068955693175 195.31776298144828
4.436687330978613 1.2674801788800205 190.63513823485545
4.436687330978613 1.5013107289081735 185.21389211820997
4.436687330978613 1.7782794100389228 179.09878813249176
4.436687330978613 2.106344542324121 172.1476395760848
4.436687330978613 2.4949326331577466 164.49799910650114
4.436687330978613 2.955209235202888 155.9952725408702
4.436687330978613 3.5003997734299794 146.74089415378523
4.436687330978613 4.146169559796819 136.77021590510637
4.436687330978613 4.911073914777701 126.30249371994579
4.436687330978613 5.817091329374358 115.4492717320483
4.436687330978613 6.890254987297223 104.48050136293467
4.436687330978613 8.16140079325183 93.5414778604823
4.436687330978613 9.66705340091042 82.94039333155744
4.436687330978613 11.450475699382825 72.87427316024377
4.436687330978613 13.562911913759416 63.51318450881763
4.436687330978613 16.065060038537286 54.96054943559491
4.436687330978613 19.028815912310247 47.296730741735786
4.436687330978613 22.539339047347912 40.508481112757444
4.436687330978613 26.697499572879334 34.567556355187094
4.436687330978613 31.622776601683793 29.416671944018667
12
5.0802180469130205 0.1 205.4666477922795
5.0802180469130205 0.11844845812380056 204.87870386947506
5.0802180469130205 0.14030037231905737 204.3603689992824
5.0802180469130205 0.16618362775387496 203.74863298805852
5.0802180469130205 0.19684194472866123 203.0271607269601
5.0802180469130205 0.23315624847200295 202.17693071288144
5.0802180469130205 0.27616998133438475 201.19188780559378
5.0802180469130205 0.32711908469136664 199.87294648692915
5.0802180469130205 0.38746751204561314 198.49178971056875
5.0802180469130205 0.45894929374867993 196.73352017318268
5.0802180469130205 0.5436183620153837 194.8401029876017
5.0802180469130205 0.6439075678850823 192.50773701977656
5.0802180469130205 0.7626985859023444 189.93391291873746
5.0802180469130205 0.9034047151333576 186.83960864620622
5.0802180469130205 1.070068955693175 183.1620002911274
5.0802180469130205 1.2674801788800205 179.0681304138075
5.0802180469130205 1.5013107289081735 174.32422693359567
5.0802180469130205 1.7782794100389228 168.97222663047214
5.0802180469130205 2.106344542324121 162.85742798700758
5.0802180469130205 2.4949326331577466 156.07726120340988
5.0802180469130205 2.955209235202888 148.52466862176828
5.0802180469130205 3.5003997734299794 140.24091844121935
5.0802180469130205 4.146169559796819 131.2996871260277
5.0802180469130205 4.911073914777701 121.75700011645145
5.0802180469130205 5.817091329374358 111.79069547049139
5.0802180469130205 6.890254987297223 101.61692435030717
5.0802180469130205 8.16140079325183 91.37134099351192
5.0802180469130205 9.66705340091042 81.35006459887414
5.0802180469130205 11.450475699382825 71.75908610078552
5.0802180469130205 13.562911913759416 62.74420617961577
5.0802180469130205 16.065060038537286 54.452796112094646
5.0802180469130205 19.028815912310247 46.95428925763593
5.0802180469130205 22.539339047347912 40.29121463970674
5.0802180469130205 26.697499572879334 34.43280486094789
5.0802180469130205 31.622776601683793 29.32967720619108
13
5.817091329374358 0.1 190.9669353070814
5.817091329374358 0.11844845812380056 190.59035372039625
5.817091329374358 0.14030037231905737 190.03621008550067
5.817091329374358 0.16618362775387496 189.5115681953148
5.817091329374358 0.19684194472866123 188.8925617641405
5.817091329374358 0.23315624847200295 188.16274807707987
5.817091329374358 0.27616998133438475 187.18371227571387
5.817091329374358 0.32711908469136664 186.1727500923907
5.817091329374358 0.38746751204561314 184.98446203210713
5.817091329374358 0.45894929374867993 183.46612669340905
5.817091329374358 0.5436183620153837 181.83439599642944
5.817091329374358 0.6439075678850823 179.824893794326
5.817091329374358 0.7626985859023444 177.5013903345675
5.817091329374358 0.9034047151333576 174.80958339428508
5.817091329374358 1.070068955693175 171.71855997141031
5.817091329374358 1.2674801788800205 168.15825768591088
5.817091329374358 1.5013107289081735 164.0160120178629
5.817091329374358 1.7782794100389228 159.32969824766658
5.817091329374358 2.106344542324121 153.9509488747759
5.817091329374358 2.4949326331577466 147.9593089053409
5.817091329374358 2.955209235202888 141.246030206115
5.817091329374358 3.5003997734299794 133.83138586549995
5.817091329374358 4.146169559796819 125.79639690616695
5.817091329374358 4.911073914777701 117.1445401507792
5.817091329374358 5.817091329374358 108.03008484137926
5.817091329374358 6.890254987297223 98.61953521552071
5.817091329374358 8.16140079325183 89.09800275207118
5.817091329374358 9.66705340091042 79.68124134169227
5.817091329374358 11.450475699382825 70.54319372149817
5.817091329374358 13.562911913759416 61.892024106300276
5.817091329374358 16.065060038537286 53.88376761390007
5.817091329374358 19.028815912310247 46.583467469823184
5.817091329374358 22.539339047347912 40.050804036726596
5.817091329374358 26.697499572879334 34.278490462902276
5.817091329374358 31.622776601683793 29.234541114195263
14
6.6608462908091575 0.1 177.494623815206
6.6608462908091575 0.11844845812380056 177.17172772225402
6.6608462908091575 0.14030037231905737 176.79023891910148
6.6608462908091575 0.16618362775387496 176.33942182139515
6.6608462908091575 0.19684194472866123 175.80733140219363
6.6608462908091575 0.23315624847200295 175.08026019282147
6.6608462908091575 0.27616998133438475 174.34131913096886
6.6608462908091575 0.32711908469136664 173.46607577076108
6.6608462908091575 0.38746751204561314 172.3511663349764
6.6608462908091575 0.45894929374867993 171.15049365147794
6.6608462908091575 0.5436183620153837 169.6415994959222
6.6608462908091575 0.6439075678850823 167.995041872159
6.6608462908091575 0.7626985859023444 165.9684741813856
6.6608462908091575 0.9034047151333576 163.6532889839832
6.6608462908091575 1.070068955693175 160.96238142868495
6.6608462908091575 1.2674801788800205 157.77541905189017
6.6608462908091575 1.5013107289081735 154.1725938868565
6.6608462908091575 1.7782794100389228 150.0514418947552
6.6608462908091575 2.106344542324121 145.406739744496
6.6608462908091575 2.4949326331577466 140.04911774885642
6.6608462908091575 2.955209235202888 134.11335580765308
6.6608462908091575 3.5003997734299794 127.51173844042565
6.6608462908091575 4.146169559796819 120.29446127691762
6.6608462908091575 4.911073914777701 112.52198300717097
6.6608462908091575 5.817091329374358 104.20685764874742
6.6608462908091575 6.890254987297223 95.55102168230745
6.6608462908091575 8.16140079325183 86.7117007398821
6.6608462908091575 9.66705340091042 77.87892661922331
6.6608462908091575 11.450475699382825 69.25130257350567
6.6608462908091575 13.562911913759416 60.98980736001808
6.6608462908091575 16.065060038537286 53.251568830376776
6.6608462908091575 19.028815912310247 46.16675485956165
6.6608462908091575 22.539339047347912 39.776134062620535
6.6608462908091575 26.697499572879334 34.10099001284812
6.6608462908091575 31.622776601683793 29.12498774620727
15
7.626985859023444 0.1 165.06074246029317
7.626985859023444 0.11844845812380056 164.78345586349567
7.626985859023444 0.14030037231905737 164.45577307658755
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16
8.733261623828433 0.1 153.58846859069297
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17
10.0 0.1 142.90197929092813
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10.0 22.539339047347912 38.7658244920976
10.0 26.697499572879334 33.4479256465557
10.0 31.622776601683793 28.7046352040252
18
11.450475699382817 0.1 133.03638684900247
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19
13.111339374215643 0.1 123.8482135621076
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20
15.013107289081734 0.1 115.36020318979611
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21
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22
19.68419447286612 0.1 100.1028186270782
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23
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24
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25
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27
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28
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29
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30
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31
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32
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76.26985859023443 2.955209235202888 46.09481860080124
76.26985859023443 3.5003997734299794 45.42414380720736
76.26985859023443 4.146169559796819 44.652251585546615
76.26985859023443 4.911073914777701 43.75651580918252
76.26985859023443 5.817091329374358 42.71941843236362
76.26985859023443 6.890254987297223 41.53333460570119
76.26985859023443 8.16140079325183 40.17944832928727
76.26985859023443 9.66705340091042 38.643947130302436
76.26985859023443 11.450475699382825 36.92793057537601
76.26985859023443 13.562911913759416 35.01880312086227
76.26985859023443 16.065060038537286 32.93067793598568
76.26985859023443 19.028815912310247 30.679868131203655
76.26985859023443 22.539339047347912 28.293657710170837
76.26985859023443 26.697499572879334 25.8198562425443
76.26985859023443 31.622776601683793 23.306180366110635
33
87.33261623828433 0.1 46.43649451457636
87.33261623828433 0.11844845812380056 46.41605766701736
87.33261623828433 0.14030037231905737 46.383857094916216
87.33261623828433 0.16618362775387496 46.355225439287736
87.33261623828433 0.19684194472866123 46.32133536653159
87.33261623828433 0.23315624847200295 46.28120058842418
87.33261623828433 0.27616998133438475 46.22525245153999
87.33261623828433 0.32711908469136664 46.16912172603966
87.33261623828433 0.38746751204561314 46.10271752186752
87.33261623828433 0.45894929374867993 46.0170959491152
87.33261623828433 0.5436183620153837 45.924270645671314
87.33261623828433 0.6439075678850823 45.80754072645235
87.33261623828433 0.7626985859023444 45.67031240778621
87.33261623828433 0.9034047151333576 45.509302998894185
87.33261623828433 1.070068955693175 45.321309911961684
87.33261623828433 1.2674801788800205 45.09427056821317
87.33261623828433 1.5013107289081735 44.830825357680204
87.33261623828433 1.7782794100389228 44.52121132793289
87.33261623828433 2.106344542324121 44.162540679427536
87.33261623828433 2.4949326331577466 43.735948233764894
87.33261623828433 2.955209235202888 43.23437595431577
87.33261623828433 3.5003997734299794 42.654706279516596
87.33261623828433 4.146169559796819 41.977274414576165
87.33261623828433 4.911073914777701 41.185266230249866
87.33261623828433 5.817091329374358 40.2785264177356
87.33261623828433 6.890254987297223 39.23126535134751
87.33261623828433 8.16140079325183 38.03379164588084
87.33261623828433 9.66705340091042 36.67075170804738
87.33261623828433 11.450475699382825 35.14109228773759
87.33261623828433 13.562911913759416 33.43432341200013
87.33261623828433 16.065060038537286 31.552977886951673
87.33261623828433 19.028815912310247 29.51005638778611
87.33261623828433 22.539339047347912 27.330708531835093
87.33261623828433 26.697499572879334 25.051600948797024
87.33261623828433 31.622776601683793 22.713341670022366
34
100.0 0.1 43.335390087438796
100.0 0.11844845812380056 43.31761501643087
100.0 0.14030037231905737 43.29658364429708
100.0 0.16618362775387496 43.265373545534274
100.0 0.19684194472866123 43.23585652230511
100.0 0.23315624847200295 43.2009670535969
100.0 0.27616998133438475 43.15280492053543
100.0 0.32711908469136664 43.103982735327094
100.0 0.38746751204561314 43.04635911656055
100.0 0.45894929374867993 42.97154701458065
100.0 0.5436183620153837 42.88362423240803
100.0 0.6439075678850823 42.788229914034176
100.0 0.7626985859023444 42.668395086396714
100.0 0.9034047151333576 42.528939362792734
100.0 1.070068955693175 42.36605781206383
100.0 1.2674801788800205 42.168408397883574
100.0 1.5013107289081735 41.93740929051012
100.0 1.7782794100389228 41.668507898570965
100.0 2.106344542324121 41.3544078787129
100.0 2.4949326331577466 40.98266242652197
100.0 2.955209235202888 40.54387163643198
100.0 3.5003997734299794 40.03602421182966
100.0 4.146169559796819 39.44242886238599
100.0 4.911073914777701 38.752987685604786
100.0 5.817091329374358 37.94960941566005
100.0 6.890254987297223 37.028083292868246
100.0 8.16140079325183 35.97004586321035
100.0 9.66705340091042 34.76592498607281
100.0 11.450475699382825 33.400570637752075
100.0 13.562911913759416 31.873827681830967
100.0 16.065060038537286 30.18646749851656
100.0 19.028815912310247 28.340994999233388
100.0 22.539339047347912 26.35725125785444
100.0 26.697499572879334 24.262362817911903
100.0 31.622776601683793 22.09530063870254
In [ ]:
np.savetxt('oligomers_a_beta-5.txt', mat_oligomers, fmt='%d')
import time
time.sleep(100)
In [ ]:
from google.colab import files

# Download the file
files.download('oligomers_a_beta-5.txt')
In [ ]:
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()
ax = fig.add_subplot(projection='3d')


x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
X, Y = np.meshgrid(x, y)  # Create a meshgrid for X and Y


ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))


ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('ln(oligomers)')
ax.view_init(elev=30, azim=120)

plt.show()
In [ ]:
In [ ]:

Protease induced by oligomers (1/P)¶

In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta/P

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])


def num_oligomers(sol):
  ts = sol[0]
  Ps = sol[3]

  cumulative_integral = [0] * len(ts)


  for i in range(2, len(ts)):
    area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
    cumulative_integral[i] = cumulative_integral[i-1] + area


  return([ts,cumulative_integral])
In [ ]:
sol = solve(m=200,beta=20)
In [ ]:
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
In [ ]:
import time
from google.colab import files
from mpl_toolkits.mplot3d import Axes3D



for beta in [20,10,5,1,0.5,0.1,0.05,0.01]:
  mnum = 35
  sigmanum = 35
  mat_oligomers = np.zeros((mnum,sigmanum))
  print(beta)
  for i,m in enumerate(np.logspace(0,2,mnum)):
    print(i)
    for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
      sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
      oligomers = num_oligomers(sol)

      final_oligomer_number = oligomers[-1][-1]

      mat_oligomers[i,j] = final_oligomer_number

  np.savetxt('oligomers_b_beta-' + str(beta) + '.txt', mat_oligomers, fmt='%d')
  time.sleep(20)
  files.download('oligomers_b_beta-' + str(beta) + '.txt')



  fig = plt.figure()
  ax = fig.add_subplot(projection='3d')


  x = np.logspace(0,2,mnum)
  y = np.logspace(-1,1.5,sigmanum)
  X, Y = np.meshgrid(x, y)  # Create a meshgrid for X and Y


  ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))


  ax.set_xlabel('m')
  ax.set_ylabel('sigma')
  ax.set_zlabel('oligomers')
  ax.view_init(elev=30, azim=120)

  plt.show()

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Protease induced by Fibrils F¶

In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta*F

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])


def num_oligomers(sol):
  ts = sol[0]
  Ps = sol[3]

  cumulative_integral = [0] * len(ts)


  for i in range(2, len(ts)):
    area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
    cumulative_integral[i] = cumulative_integral[i-1] + area


  return([ts,cumulative_integral])
In [ ]:
sol = solve(m=5000000,beta=100)
In [ ]:
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([50,200])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with $\beta = 100$ and $m=5000000$")
plt.show()
In [ ]:
import time
from google.colab import files
from mpl_toolkits.mplot3d import Axes3D



for beta in [0.005]:
  mnum = 35
  sigmanum = 35
  mat_oligomers = np.zeros((mnum,sigmanum))
  print(beta)
  for i,m in enumerate(np.logspace(0,2,mnum)):
    print(i)
    for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
      sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
      oligomers = num_oligomers(sol)

      final_oligomer_number = oligomers[-1][-1]

      mat_oligomers[i,j] = final_oligomer_number

  np.savetxt('oligomers_c_beta-' + str(beta) + '.txt', mat_oligomers, fmt='%d')
  time.sleep(20)
  files.download('oligomers_c_beta-' + str(beta) + '.txt')



  fig = plt.figure()
  ax = fig.add_subplot(projection='3d')


  x = np.logspace(0,2,mnum)
  y = np.logspace(-1,1.5,sigmanum)
  X, Y = np.meshgrid(x, y)  # Create a meshgrid for X and Y


  ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))


  ax.set_xlabel('m')
  ax.set_ylabel('sigma')
  ax.set_zlabel('oligomers')
  ax.view_init(elev=30, azim=120)

  plt.show()

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In [ ]:

Read data and make graphs¶

In [ ]:
import numpy as np
import matplotlib.pyplot as plt

import numpy as np

# Mount Google Drive
from google.colab import drive
drive.mount('/content/drive')

Mounted at /content/drive

Normal model¶

In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve_original_model(gamma=1,m=25,sigma=1,nu=1,
                         ic = [0,0,0], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
In [ ]:
sol_m25 = solve_original_model(m=25)


plt.plot(sol_m25[0], sol_m25[1],label="F(t)",color="blue")
plt.plot(sol_m25[0],sol_m25[2],label="C(t)",color="green")
plt.plot(sol_m25[0],sol_m25[3],label="P(t)",color="red")

plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
In [ ]:
from mpl_toolkits.mplot3d import Axes3D
mnum = 35
sigmanum = 35

def plot_matrix(file_path,title="",limi=14):
  mat_oligomers = np.loadtxt(file_path, delimiter=' ')
  fig = plt.figure()
  ax = plt.axes(111, projection='3d')

  x = np.logspace(0,2,mnum)
  y = np.logspace(-1,1.5,sigmanum)

  X, Y = np.meshgrid(x, y)  # Create a meshgrid for X and Y

  ax.plot_wireframe(X,Y,np.log(mat_oligomers.T),color="navy")


  ax.set_xlabel('m')
  ax.set_ylabel(r'$\sigma$')
  ax.set_zlabel('ln (oligomer )')

  ax.view_init(elev=30, azim=100)

  plt.title(title)
  ax.set_zlim([2,limi])

  plt.show()
In [ ]:
plot_matrix("/content/drive/MyDrive/Project/oligomers_normal.txt", "Original result")

Induced by protofibrils¶

$$ \dfrac{dP}{dt} = \sigma - P -\gamma F \cdot P + \nu C + \beta \dfrac{m}{1+P} $$
In [2]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta*m/(1+P)

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
In [14]:
sol = solve(m=20000,beta=0.4,sigma=1,tmax=1000,ic=[2.5,250,90])

plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([800,1000])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution")
plt.show()
In [ ]:
for beta in ["0.025","0.05","0.1","0.2","0.5","1","5"]:
  plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_a_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)
In [ ]:

Induced by oligomers¶

$$ \dfrac{dP}{dt} = \sigma - P - \gamma F \cdot P + \nu C + \beta \dfrac{1}{P} $$
In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta/P

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
In [ ]:
sol = solve(m=25,beta=0.1)

plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
In [ ]:
for beta in ["0.01","0.05","0.1","0.5","1","10","20"]:
  plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_b_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)

Induced by Fibrils¶

$$ \dfrac{dP}{dt} = \sigma - P - \gamma F \cdot P + \nu C + \beta F $$
In [ ]:
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt

def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
                         ic = [0,0,0], t0 = 0, tmax=200,start_from=50):


  def derivs(t,var):
    dvar = np.zeros(3)
    F,C,P = var
    dvar[0] = m/(1+P) - gamma * F*P
    dvar[1] = gamma*F*P - nu*C
    dvar[2] = sigma - P - gamma*F*P + nu*C + beta*F

    return dvar

  sol = solve_ivp(derivs,[t0,tmax],ic)

  ind = list(sol.t>start_from).index(True)


  return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
In [ ]:
sol = solve(m=25,beta=0.1)

plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")

plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
In [ ]:
for beta in ["0.005","0.01","0.02","0.05 (1)","0.1","0.5","1","5","10"]:
  plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_c_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)
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